Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


this topology, is called the orbit space of the action of G on E. The mapping
Ui-7i~1(U) is a bijection of the set of open sets of E/G onto the set of satura-
ted open sets of E. A subset F of E/G is closed in E/G if and only if n~ 1(F) is
closed in E, because /T^E/G) - F) = E - n~l(F).

(12.10.6)   (i)   The canonical mapping n:E-+ E/G is continuous.
(ii)   The image under n of every open set in E is open in E/G.
(iii)   A mapping f of E/G into a topological space E' is continuous if and
iff0 n : E -> E' is continuous.

The first assertion follows from the definition of the topology of E/G and
from (3.1 1 .4(b)). To prove the second, it is enough to show that if V is open in
E then its saturation G  V = n~1(n(V)) is open in E; and this is clear because
G  V = U s  V, and each s  V is open in E by virtue of (12.10.3). Finally, if


/: E/G-+E' is continuous, then so is/o n by (i); conversely, if/o n is con-
tinuous, then for every open set U' in E' the set n'i(f~1(U/J) is open in E,
hence /-1(U') is open in E/G. This proves (iii).

If x is any point of E and if V runs through a fundamental system of
neighborhoods of x in E, the sets rc(V) form a fundamental system of neigh-
borhoods of the point n(x) in E/G.

(12.10.7)    Let A be a subset of E and let A' = G - A = TT^A)) be its
saturation with respect to G. Then the canonical bijection q> of the subspace
7t(A) of E/G onto the orbit space A'/G is a homeomorphism.

The mapping (p is defined as follows : if x e A, the image under cp of the
orbit G  x is the same orbit considered as an element of A'/G. As U runs
through the set of open subsets of E/G, the mapping cp takes U n 7i(A) to
the canonical image ofn~1(U) n A' in A'/G. Now U n n(A) runs through the
set of open subsets of the subspace 7i(A) of E/G, and q>(n~1(U) n A') runs
through the set of open subsets of A'/G. For if V is a saturated open sub-
set of -the subspace A' of E, then V7 is of the form V n A', where V is open
in E; but also V = (G  V) n A' = rc' ^(V)) n A', and 7t(V) is open in E/G
by (12.10.6).

(12.10.8)    Let G be a topological group acting continuously on a topological
space E. In order that E/G should be Hausdorff it is necessary and sufficient
that, in the product space E x E, the set R of pairs (x, y) belonging to the same
orbit be closed. When this is the case, every orbit is closed in E. The orbits are {0} and